Problem: You have found the following ages (in years) of 4 turtles. Those turtles were randomly selected from the 48 turtles at your local zoo: $ 46,\enspace 67,\enspace 105,\enspace 57$ Based on your sample, what is the average age of the turtles? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we only have data for a small sample of the 48 turtles, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\overline{x}} = \dfrac{46 + 67 + 105 + 57}{{4}} = {68.8\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {519.84} + {3.24} + {1310.44} + {139.24}} {{4 - 1}} $ {s^2} = \dfrac{{1972.76}}{{3}} = {657.59\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{657.59\text{ years}^2}} = {25.6\text{ years}} $ We can estimate that the average turtle at the zoo is 68.8 years old. There is also a standard deviation of 25.6 years.